Abstract
Pump impellers and pump casings are considered as simultaneous beams coupled through the bearings, the stuffing-box and the radial clearances acting as hydrostatic bearings.
Lomakin has given a relation for the restoring forces in smooth clearances, according to which these are to be taken as springy supporting bearings the spring constants of which are proportional to the pressure. Small clearances therefore raise the critical speed.
The procedure is a method of residues in which the estimated natural frequency is varied until the residual value diminishes. In the case of multiple masses and supports the calculation necessitates the use of digital computers.
The critical speed is calculated by extending the matrix-method known from the single beam to systems of two parallel beams. Because of the analogy of the differential equations for the shaft operating with critical speed and for the transversally vibrating beams, rotation of one beam is allowed also. By special transposition matrices, single masses, spring-elastic couplings between the two beams may be considered. The coupling forces according to Lomakin, already explained above, being proportional to the product from deflection and square of speed, formally correspond to the loading of the shaft by apparent negative masses. In the case of very close clearances the critical speed may be almost completely suppressed.
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